Abelian variety isogeny class 2.113.abj_ug over $\F_{113}$

• Label: 2.113.abj_ug
• Base field: $\F_{113}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{113}$
Dimension:	$2$
L-polynomial:	$( 1 - 20 x + 113 x^{2} )( 1 - 15 x + 113 x^{2} )$
	$1 - 35 x + 526 x^{2} - 3955 x^{3} + 12769 x^{4}$
Frobenius angles:	$\pm0.110150159186$, $\pm0.250704227710$
Angle rank:	$2$ (numerical)
Jacobians:	$30$
Isomorphism classes:	130

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9306$	$160863516$	$2082659572224$	$26587833356246976$	$339461030529394093386$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$79$	$12597$	$1443388$	$163068161$	$18424584719$	$2081953111698$	$235260549097583$	$26584441906666753$	$3004041939047572684$	$339456739028028581397$

Jacobians and polarizations

This isogeny class contains the Jacobians of 30 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=76x^6+7x^5+110x^4+62x^3+96x^2+18x+62$
• $y^2=103x^6+75x^5+105x^4+97x^3+38x^2+43x+7$
• $y^2=31x^6+41x^5+79x^4+77x^3+51x^2+24x+94$
• $y^2=67x^6+13x^5+60x^4+51x^3+33x^2+45x+35$
• $y^2=59x^6+95x^5+18x^4+57x^3+39x^2+77x+12$
• $y^2=24x^6+70x^5+36x^4+51x^3+44x^2+85x+33$
• $y^2=67x^6+23x^5+106x^4+31x^3+107x^2+81x+27$
• $y^2=32x^6+8x^5+19x^4+43x^3+96x^2+70x+66$
• $y^2=101x^6+25x^5+50x^4+35x^3+44x^2+20x+73$
• $y^2=24x^6+23x^5+64x^4+70x^3+72x^2+63x+99$
• $y^2=63x^6+61x^5+17x^4+105x^3+36x^2+104x+79$
• $y^2=46x^6+103x^5+14x^4+64x^3+112x^2+36x+57$
• $y^2=78x^6+53x^5+19x^4+73x^3+94x^2+57x+89$
• $y^2=2x^6+63x^5+93x^4+56x^3+50x^2+64x+28$
• $y^2=102x^6+72x^5+88x^4+23x^3+74x^2+35x+31$
• $y^2=48x^6+9x^5+54x^4+97x^3+81x^2+80x+24$
• $y^2=107x^6+95x^5+9x^4+99x^3+56x^2+105x$
• $y^2=67x^6+85x^5+6x^4+66x^3+7x^2+40x+3$
• $y^2=12x^6+35x^5+43x^4+53x^3+97x^2+109x+75$
• $y^2=107x^6+90x^5+85x^4+103x^3+26x^2+79x+55$
• $y^2=46x^6+41x^5+71x^4+42x^3+24x^2+78x+58$
• $y^2=43x^6+104x^5+54x^4+83x^3+49x^2+81x+32$
• $y^2=21x^6+100x^5+83x^4+51x^3+33x^2+67x+103$
• $y^2=21x^6+14x^5+9x^4+18x^3+44x^2+61x+12$
• $y^2=66x^6+37x^5+38x^4+55x^3+66x^2+3x+74$
• $y^2=96x^6+92x^5+57x^4+88x^3+26x^2+91x+6$
• $y^2=9x^6+71x^5+6x^4+40x^3+91x^2+93x+74$
• $y^2=107x^6+87x^5+12x^4+50x^3+17x^2+41x+6$
• $y^2=94x^6+73x^5+85x^4+80x^3+88x^2+25x+72$
• $y^2=50x^6+52x^5+26x^4+92x^3+60x^2+43x+51$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{113}$.

Endomorphism algebra over $\F_{113}$

The isogeny class factors as 1.113.au $\times$ 1.113.ap and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.113.au : \(\Q(\sqrt{-13}) \).
• 1.113.ap : \(\Q(\sqrt{-227}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.113.af_acw	$2$	(not in LMFDB)
2.113.f_acw	$2$	(not in LMFDB)
2.113.bj_ug	$2$	(not in LMFDB)